The nodes or abscissaszk are real or complex; function values
are fk=f⁢(zk). Given n+1 distinct points zk and n+1 corresponding
function values fk, the Lagrange interpolation polynomial is the
unique polynomial Pn⁡(z) of degree not exceeding n such that
Pn⁡(zk)=fk, k=0,1,…,n. It is given by

The interpolation error Rn⁡(z) is as in §3.3(i). Newton’s
formula has the advantage of allowing easy updating: incorporation of a new
point zn+1 requires only addition of the term with
[z0,z1,…,zn+1]⁢f to (3.3.38), plus the computation
of this divided difference. Another advantage is its robustness with respect to
confluence of the set of points z0,z1,…,zn. For example, for k+1
coincident points the limiting form is given by
[z0,z0,…,z0]⁢f=f(k)⁢(z0)/k!.

§3.3(v) Inverse Interpolation

In this method we interchange the roles of the points zk and the function
values fk. It can be used for solving a nonlinear scalar equation f⁢(z)=0
approximately. Another approach is to combine the methods of §3.8
with direct interpolation and §3.4.

and with f=0 we find that x=-2.33823 2462, with 4 correct digits. By
using this approximation to x as a new point, x3=x, and evaluating
[f0,f1,f2,f3]⁢x=1.12388 6190, we find that x=-2.33810 7409, with
9 correct digits.

and compute an approximation to a1 by using Newton’s rule
(§3.8(ii))
with starting value x=-2.5. This gives the new point
x3=-2.33934 0514. Then by using x3 in Newton’s interpolation
formula, evaluating [x0,x1,x2,x3]⁢f=-0.26608 28233 and
recomputing f′⁢(x), another
application of Newton’s rule with starting value x3 gives the
approximation x=2.33810 7373, with 8 correct digits.